    Canceling Upper and Lower Surface Boundary Layer Vorticity at the Trailing Edge
This invention relates to methods of boundary layer control (BLC) on a finite lifting body to cancel boundary layer vorticity as it sheds from the trailing edge. These methods do not contradict Hemholt's theorems which state that vorticity cannot be terminated within an ideal fluid. The reason is that viscosity in real fluids allows boundary layer vorticity to be removed by suction into the vacuum side of a mechanical blower or ejector, where fluid viscosity dissipates its vorticity. This process is often displayed in museums when simulating a tornado. A Plexiglas cylinder is sealed on the bottom. A pan with boiling water, or smoke generator is placed in its middle. A small exhaust fan mounted in the middle of the top cover is started up. As air is drawn out of the cylinder, fresh air is allowed to enter through tangential entry ports into the cylinder. Conservation of angular momentum increases its angular velocity as it approaches the centerline and there the pressure drops. The radial pressure gradient next to the floor is reduced by friction thereby setting up a radial inflow teacup like flow pattern. Steam or smoke released in the middle renders this simulated tornado visible. The fan in the exhaust completely dissipates the entering tornado vorticity by viscous effects in its rotor and none of it remains in the exhaust.
Therefore, five theorems for real viscous fluids are formulated here.    1) To produce lift on an airfoil, a viscous boundary layer is required. The lower surface boundary layer satisfies the Kutta condition at the trailing edge and provides entrainment for the upper boundary layer to do likewise.    2) Vorticity is generated throughout the boundary layer of thickness δ by shear forces. A contour integral taken clockwise in the plane perpendicular to the span, and external to the boundary layer of thickness δ, equals the bound circulation Γ.    3) The boundary layer vorticity in the upper and lower boundary layer are in opposite directions. Only when their integral value is different in magnitude will the contour integral Γ={right arrow over (V)}·{right arrow over (dl)}, produce a net bound circulation Γ, which determines lift and induced drag per unit span: {right arrow over (L)}=ρ∞({right arrow over (V)}∞×{right arrow over (Γ)}) and {right arrow over (D)}i=ρ∞({right arrow over (w)}1×{right arrow over (Γ)}).    4) A uniform finite 3-D wing, can only have a uniform span wise loading if equipped with:    a) A wing tip vortex countering device, capable of eliminating all span wise pressure gradients so that the bound circulation vector will align with the wing span all the way down to the wing tip. Only then will the vorticity in upper and lower boundary layer cancel each other upon shedding from the trailing edge.    b) To assist in the cancellation of boundary layer vorticity upon shedding, a short suction slot should be added to the trailing edge adjacent to the wing tip.    5) Consider a wind tunnel test, with a uniform wing mounted in between two opposing walls, and with tunnel wall-boundary layer removed by suction through pores in the wind tunnel walls in order to assure uniform velocity over the wing. To make the wing bound circulation truly two-dimensional (2-D), the boundary layer formed in the corners between the wing and the wall must also be removed by suction. This is accomplished by adding a narrow suction slot in the upper and lower surface corner intersection with the walls.The success of achieving near 2-D loading on a finite wing depends on the wing tip vortex-countering device used. This can range from an end plate, with or without blowing, or by a propeller or turbine rotor producing a wake with sufficient counter rotation. If this is over-driven, the result will be an induced updraft on the wing, producing some induced thrust instead of induced drag, which compensates partially for the excess power used to over-drive the counter rotating vortex. When successful in preventing shedding vorticity into the wake, and thus eliminating induced drag at all flight speeds V∞, the ideal thrust-power required TPideal can be related to that of a wing with elliptic loading by:
            TP      ideal              TP      elliptic        =                              C          Dpara                *                  (                      0.5            ⁢                                                  ⁢                          ρ              ∞                        ⁢                          S              w                                )                *                  V          ∞          3                                      C          Dpara                *                  (                      0.5            ⁢                                                  ⁢                          ρ              ∞                        ⁢                          S              w                                )                *                  V          ∞          3                *                  [                                                    (                                                      V                    opt                                                        V                    ∞                                                  )                            2                        +            1                    ]                      =          1              [                                            (                                                V                  opt                                                  V                  ∞                                            )                        2                    +          1                ]            This shows that thrust power savings can exceed 50% when V∞<Vopt, but are less than 50% when V∞>Vopt.
The following is an estimate of the power required for boundary layer by suction at V∞=Vopt, to prevent vorticity from shedding from the trailing edge of a uniformly loaded wing. To provide boundary layer removal by suction all along the trailing edge, as would be needed with a elliptically loaded wing would require far more power then possibly saved in thrust power! This analysis is based on assuming an aircraft flying at a Reynolds number Re=106 and (L/D)max=15 at CL=0.6, aspect ratio AR=10 and trailing edge turbulent boundary layer thickness δ=0.37*c/(Re)0.2=0.023*c. For all boundary layers, with thickness δ, one can assume that most of the vorticity is contained within its momentum displacement thickness δ2≅0.1*δ. Consider removing the boundary layer vorticity by suction at the trailing edge over a distance of ½c from the wing tip. This requires removing at least a total volume flow rate of: Qsuction=4*(½c)*(δ2)*V∞=2*c2*0.0023V∞, assuming a suction Δp equal to −q28 , then the minimum suction power required, Psuction=q∞*0.0046*c2*V∞. Comparing to the thrust power required Pthrust=D*V∞=(L/(L/D))*V∞=CL*q∞*(Sw=AR*c2)*V∞/(L/D), the ratio of Psuction/Pthrust=0.0046/(CL*AR/(L/D)=0.012.
If this suction power is supplied by small wind turbines, one on each wing tip at an assumed 25% efficiency, the ratio of suction power to thrust power required Psuction/Pthrust=0.012/0.25=4.28%. The wing tip mounted wind turbine wake circulation may be sufficient to eliminate span-wise pressure gradients near the wing tip.
An alternate solution is to use an ejector to provide the required suction power and using its exhaust to provide Coanda blowing over a rounded wing tip making the associated power required higher. Make the following assumptions: provide a suction at Δp=−q∞. Use a Coanda jet blowing velocity of at least Vjet=1.5 V∞ or a dynamic pressure qjet=1.52 q∞; supply air ejector with volume flow rate of Qsupplied=Qsuction; assume an ejector efficiency of 50%. Equating the compressed air supply power minus the power required for ejector suction to the blowing air power required: 50%*(qsupplied*Qsupplied)−(q∞*Qsuction)=qjet*(Qsupplied+Qsuction). The compressed air supply pressure is: qsupplied=11q∞. In that case, Psuction/Pthrust=11*0.008=9%. If this technique eliminates the induced drag power when flying at Vopt, then the thrust power savings are still 50%−9%=41% (four times better than achievable with a winglet).
The savings in cruise power promise to be significant. Increase in power available during take-off and landings, with a high lift system, will be even more significant. This will eliminate the current problem of having to operate on the backside of the power curve with most high lift systems.
Additional benefits of having a uniformly loaded rectangular wing with BLC and same root chord and wing area are that it reduces wingspan by 21.5% and the wing root bending moment by 7.5%. By eliminating wing twist the average lift coefficient increases allowing for a higher payload. This would allow a new large airplane with 510 passengers, a range of 8000 nautical miles and a 650 square meter wing area to have the same wingspan and root chord as the current 400 passengers Boeing 747–400 with 511 square meter wing area.
At low speed it also provides a reliable technology for wing tip stall protection. Then deflecting both ailerons upwards will reduce their effective angle of attack. Roll control can also be achieved by asymmetric reduction in boundary layer suction and blowing. This produces an induced drag with loss of lift resulting in a coordinated turn, which is better than conventional ailerons with their adverse roll-yaw coupling and tip stall characteristics. This same technology can be applied to helicopter rotors to replace cyclic pitch control and reduce noise on propeller and fan blades as well. Then the required BLC compressed air can be provided by a pumping action inside its hollow blades.
The theories presented herein are meant to provide aerodynamic background for this invention only and are not intended to limit the scope of the invention or to serve as the only means by which the embodiments of the invention may operate.